Multiplicative Updates for Convolutional NMF Under $β$-Divergence

AbstractWe generalize the convolutional NMF by taking the $$\beta $$β-divergence as the contrast function and present the exact multiplicative updates for its factors in closed form. The new updates unify the $$\beta $$β-NMF and the convolutional NMF. We state why almost all existing updates are inexact and/or approximative w.r.t. the convolutional data model. In addition, we prove that the $$\beta $$β-divergence is nonincreasing under our updates and confirm numerically that the updates are stable and that the convergence of the contrast function is consistent across the most common values of $$\beta $$β.